Small-Signal Stability Analysis of Distorted Power Systems installed with SSSC

M. Ladjavardi, Student Member, IEEE M.A.S. Masoum, Senior Member, IEEE S.M. Islam, Senior Member,IEEE marjan.ladjavardi@postgrad.curtin.edu.au m.masoum@curtin.edu.au s.islam@ece.curtin.edu.au

Department of Electrical and Computer Engineering, Curtin University of Technology,

Perth, Western Australia

ABSTRACT Time and space harmonics can cause variations on synchronous generator steady state operating conditions and the small-signal stability of power system. Static Synchronous Series Compensator (SSSC) is a FACTS device which is widely used in transmission lines. In this paper eigen-value analysis method is used to investigate the impact of time and space harmonics on small signal stability of Single Machine Infinite Bus (SMIB) system installed with SSSC. System state space equations are calculated considering additional terms due to the presence of harmonics. Simulation results are demonstrated for different harmonic levels and show the considerable impact of harmonics on the system eigen-values and hence dynamic behavior of the system. Index Terms— Synchronous machine, small-signal stability, SSSC, load angle, time and space harmonics.

1. INTRODUCTION

There are many nonlinear devices that generate and inject harmonic currents and voltages into power system. This calls for the power system stability studies to be explored in non-sinusoidal operating environments. Harmonic pollutions in power system tend to initiate and/or amplify stability problems. Flexible AC Transmission Systems (FACTS) are recognized as a transmission transfer capacity enhancement solution [1-11]. Static Synchronous Series Compensator (SSSC) is a FACTS device to be connected in series with power transmission lines. With the injected voltage in quadrature with the line current and the capability of dynamically changing its reactance characteristic from capacitive to inductive, SSSC has become very effective for power flow control [12,13]. Presence of time and space harmonics can cause variation on the induced fundamental voltage of synchronous generator (SG). Therefore, accurate modeling of SG is needed to compute the initial steady state operating conditions considering time and space harmonics.

The literature is rich in linear and nonlinear models of SM for steady-state and transient studies. SM is modeled using different frames of reference including dqo and αβo-coordinates in time or frequency domain [14-29]. A nonlinear model of SG using abc-frame and Harmonic Domain (HD) is used to calculate the load angle and investigate the small signal stability of the system in the presence of time and space harmonics [29-30]. According to these references, time and space harmonics modify the fundamental stator flux and will generate additional components. Therefore, Park transformation and eigen-value analysis are used to investigate the effect of time and space harmonics on small-signal stability of SG and power system [30]. This paper uses the nonlinear model of SG (developed in [30]) to investigate the stability of distorted power systems. To investigate the effect of time and space harmonics on small-signal stability of SG and power system the eigen-value analysis is performed [31]. Installation of FACTS devices will affect the stability of the system. In this paper SSSC is assumed to work on the fundamental frequency without additional power oscillation damper and its open loop dynamic model is included in the system state matrix. Considering additional terms of the stator fluxes due to harmonics on d-q axis, the general method to calculate the eigen-values of a Single Machine Infinite Bus (SMIB) system including SSSC is obtained. Simulation results for a SMIB system show variations in the eigen-values of the “A” matrix. Impact of harmonic (magnitudes and phase angles) variations on the steady-state operating condition and eigen-values of the system are observed.

2. SYNCHRONOUS GENERATOR MODEL FOR NON-SINUSOIDAL OPERATION

Synchronous generator (SG) is modeled in harmonic domain (HD) using abc-frame of reference [28-30]. Park transformation is applied to determine the induced fundamental voltages. This model includes space harmonics and rotor saliency.

1-4244-1298-6/07/$25.00 ©2007 IEEE.

Considering the steady state performance of the SG in HD, the main generator equations are as follows:

⎪⎩

⎪⎨

⎧

+=+=

+=−−

−−−

rsrssss

srsrrrrrr

rsrrrrrrr

ILiL

ILLLI

ULLRLRD

][][

][][

][][])[]([11

111

ψψ

ψ (1)

where the all matrixes are defined in [30]. Equation 1 shows that the Stator flux includes fundamental and harmonic components with constant magnitudes that rotate at different speeds with respect to the rotor. The fundamental component of stator flux (generated by the interaction of fundamental and harmonics stator current and inductances) can be written as follows.

)1(harmtodues

)0(r

)1(sr

)1(s

)0(ss

)1(s

)2(ss

)1(s iLiLiL −+++= ψψ (2)

where under sinusoidal operating conditions, the additional terms due to time and space harmonics ( )1(

harmtodues−ψ ) will be zero. Based on equation 1, stator

time harmonics will induce current harmonics in rotor, which may affect stator fluxes. Therefore, damper windings can be effective in the steady state operation of synchronous generators in the presence of harmonics.

The modified fundamental stator flux with harmonic consideration in dq0-coordinates is obtained using Park transformation [30]:

harmtodueq

)1(qqq

harmtodued

)0(F

)1(F

)1(ddd

ψiLψ

ψIMiLψ

+−=

++−= (3)

harmtoduedψ and harmtodue

qψ are stator fluxes resulting from

interaction between time and space harmonics that depend on the initial rotor angle (

oθ ). The variables of

equation 3 are defined in [30]. For stability studies, it is necessary to estimate the initial steady-state load angle of synchronous generator as a function of non-sinusoidal terminal quantities. The generator load (rotor) angle in the presence of time and space harmonics is computed as follows [30].

)sin()sin()cos(

)cos()sin()cos(

0)1()1()1(

0)1()1(

0 δψφφδψφφ

δharmtodue

qtqtat

harmtodueqtatq

ILIRE

IRIL

+++−−

=

(4) where φ is the power factor angle. According to Eq. 4,

the rotor angle will vary with magnitude and phase of injected time harmonics in a generator with specific space harmonics. Note that under sinusoidal (harmonic-free)

operating conditions, harmtodueqψ will be equal to zero.

3. SYSTEM INVESTIGATED

A distorted Single Machine Infinite-Bus (SMIB) power system installed with SSSC is investigated, as shown in figure 1.

Figure 1.The distorted single-machine infinite-bus (SMIB) power system installed with a SSSC, used for the analysis.

The SSSC consists of a boosting transformer with a

leakage reactance of ssscX , a three-phase GTO based

voltage source converter ( INVV ) and a DC capacitor

( DCC ). Signal m is the amplitude modulation ratio of the

Pulse Width Modulation (PWM) based Voltage Source Converter (VSC), which is the input control signal to the SSSC and determines the magnitude of the inserted voltage. Also, signal ψ is the phase of the injected

voltage and is kept in quadrature with the line current (inverter losses are ignored). Therefore, the compensation level of SSSC can be controlled dynamically by changing the magnitude of the injected voltage. Time harmonics are assumed to be injected by a nonlinear load (presented by decoupled harmonic current sources) and are constant during the perturbation. Also, the effect of load angle variations on the induced rotor harmonic currents is ignored.

3.1 STATE SPACE EQUATIONS

The eigen-value analysis method is used to investigate the impact of harmonics on the small signal stability of the SMIB system (Fig. 1).

The system state space equations have been calculated in the following general form [14]:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆

∆∆

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

∆

∆∆∆

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

∆

∆∆

∆

• m

E

T

B

V

A

V

fd

m

DC

fd

r

DC

fd

r

ψδω

ψδω

&

&

&

(5)

where matrices A and B are calculated using the linearized swing equation, the field circuit and SSSC differential equations around a nominal operating point. The differential equations corresponding to the Synchronous generator and SSSC are as follows:

o90

)sincos(

)(

)(2

1

)1()1(

0

0

)1(

±=

+=

∠=

−=

∆=

∆−∆−∆=∆

φβ

ββ

β

ωψ

ωωδ

ωω

qdDC

DC

DCINV

FdFdFdFd

r

rDemr

iiC

mk

dt

dV

mkVV

iRedt

ddt

d

KTTHdt

d

(6)

where k is the ratio between AC and DC voltages and depends on the inverter structure and φ is the power

factor. The SSSC model may not be valid for transient phenomena and/or unsymmetrical operating conditions [7-9].

)1(eT (the produced electrical torque at synchronous speed

by fundamental and harmonic currents and voltages) is computed using the following equation [14]:

)1()1()1()1()1(dqqde iiT ψψ −= (7)

)1(dψ and

)1(qψ can be computed using equation 3. To

computed )1(qi and )1(

di , system and machine equations

are written for the fundamental frequency.

System equations:

)1()1()1( ))((~~

IXXjREVE ssscEEBINVt ++++=

⎪⎩

⎪⎨⎧

++++=

+−++=)1()1()1()1(

)1()1()1()1(

)(

)(

dssscEqEINVqBqq

qssscEdEINVdBdd

iXXiRVEe

iXXiRVEe (8a)

Machine equations:

⎪⎩

⎪⎨⎧

+−=

−−=)1()1()1(

)1()1()1(

dqaq

qdad

iRe

iRe

ψ

ψ (8b)

where,

⎪⎩

⎪⎨⎧

=

=

⎪⎩

⎪⎨⎧

=

=

)(cos

)(sin,

)(cos~

)(sin~

)1()1(

)1()1(

)1()1(

)1()1(

δδ

δδ

BBq

BBd

itq

itd

EE

EE

Ee

Ee

Using 8a and 8b, the fundamental stator currents are obtained:

2

2

)1(

)1(

2

2

)1(

)1(

)cos(

]sin)([

)cos(

]sin)([

TTqdTq

TdDCBdharmtodue

q

TTqdTq

DCharmtodue

dBqFd

FFdT

q

TTqdTq

TDCBdharmtodue

q

TTqdTq

DCharmtodue

dBqFd

FFdTq

d

RXX

XmkVE

RXX

mkVEL

MR

i

RXX

RmkVE

RXX

mkVEL

MX

i

+++

++

−+−=

+++

−+

−+−=

βψ

βψψ

βψ

βψψ

aET

Fd

FdssscETd

qssscETq

RRR

L

MLXXX

LXXX

+=

−++=

++=)1(

(9)

In order to identify system eigen-values, the electrical torque is linearized around the operating point:

)1()1(0

)1(0

)1()1()1(0

)1(0

)1()1(dqdqqdqde iiiiT ∆−∆−∆+∆=∆ ψψψψ (10)

The initial values for stator current and fluxes on d-q axis can be calculated using equations 3, 4 and 9. Also, from equations 3 and 9, derivations of stator currents and fluxes on d-q axis are carried in the following general form:

⎪⎩

⎪⎨

⎧

=∆+∆

+∆+∆+∆+∆=∆

)11(,,,65

4321

qdqd

DC

harmtodueq

harmtoduedFd

iif

maVa

aaaaf

ψψ

ψψδψ

Using [30], harmtoduedψ∆ and harmtodue

qψ∆ will be

determined as a function of iθ∆ , which is equal to o

θ for

steady state operations.

From equation (8):

])(

)([arctan

)1()1()1(

)1()1()1(

dssscEqEINVqBq

qssscEdEINVdBdi iXXiRVE

iXXiRVE

+++++−++

=δ (12)

and,

ii δθ ∆=∆ .

Therefore, the derivation of the stator fluxes produced by the interaction between space and time harmonics ( harmtodue

qharmtodue

d and ψψ ∆∆ ) are calculated as a function

of state space variables in following form:

mbVbbb DCfdharmtodue

qd ∆+∆+∆+∆=∆ 4321, ψδψ (13)

Substituting harmtoduedψ∆ and harmtodue

qψ∆ in equations

11, derivation of electrical torque can be computed in the following general form:

mKVKKKT DCFde ∆+∆+∆+∆=∆ 4321)1( ψδ (14)

where 4321 KandK,K,K are functions of initial stator

(fundamental and time harmonics) currents, initial stator fluxes, steady state load angle, rotor (dc and harmonic components) currents, DC capacitor voltage, modulation ratio of the PWM based VSC and space harmonics.

Using equation 6 and substituting the values of currents from equation 11, the derivations of DC capacitor voltage and field flux can be calculated as follows:

mNVNNN DCFdfd ∆+∆+∆+∆=∆•

4321 ψδψ

mMVMMMV DCFdDC ∆+∆+∆+∆=∆•

4321 ψδ (15)

The coefficients in the above equation are also functions of the initial operating conditions.

Finally, using equations 14 and 15, the system state space equation will be found as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆

∆∆

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡ −

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

∆

∆∆∆

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡ −−−−

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

∆

∆∆

∆

•

m

E

T

M

NM

R

H

K

H

VMMM

NNN

H

K

H

K

H

K

H

K

V

fd

m

F

fd

DC

fd

rD

DC

fd

r

.

00

0

0002

02

1

.

0

0

0002222

3

4)1(

0

4

321

321

0

321

ω

ψδω

ωψδω

&

&

&

(16)

4. SIMULATION RESULTS

To illustrate the impact of time and space harmonics on small-signal stability of the power system, variations of the steady state operating conditions and eigen-values of the system state matrix (A) are presented. The system under study is a distorted power system which includes a SG injecting space harmonics, a nonlinear load injecting time harmonics, an infinite bus and a SSSC unit which injects voltage at the fundamental frequency to the transmission line (Fig.1). Specifications of the synchronous generator, power system and SSSC are provided in Appendix A [1]. Variations of eigen-values (compared with the sinusoidal operation) are used to show the impact of harmonics on the dynamic operation of the system. Case 1- Sinusoidal Operation:

The effect of space harmonics is neglected and stator current is assumed to be sinusoidal. Synchronous

generator is delivering 555 MVA at 0.9 p.f. (lag) with rated terminal voltages. The effect of magnetic saturation and damper windings are neglected. Using equations 3, 4, 9 and 16, the steady state operating values and system eigen-values are calculated as follows:

=0δ 39.0757, =0dI 0.9058, =0qI 0.4239,

=0dψ 0.7778, =0qψ -0.6332

⎩⎨⎧

±±

=0.0283j 0.0162-

5.1993j 0.1473-λ

Case 2- Non-Sinusoidal Operation in presence of Time and Space Harmonics:

To include the effects of time and space harmonics, similar harmonic orders and levels as references [30] are included. Space harmonics are close to the typical values as suggested in reference [29]. It is sufficient to include even space harmonics in the self-stator inductance [29,30] and consider only odd harmonics in the mutual stator-rotor inductance [29,30]. Current harmonic phase angles will also be included since they have impacts on the voltage and flux wave shapes and may influence the stability of the system. In this case, the self-stator inductance has additional even components (up to the 10th harmonic) and the mutual stator-rotor inductance contains odd components (up to the 9th harmonic). The selected base values for space harmonics are 0.3, 0.2, 0.1 and 0.01 of the second and the first harmonic for stator and stator-rotor and the base values for time harmonics are 0.3pu, 0.5pu, 0.2pu and 0.1pu for the 3rd to 9th harmonics, respectively. Loading condition is the same as case 1. To show the effect of current harmonic phase angles, two different values (0 and π/2) for the 7th harmonic are considered at each (time) harmonic level. Tables 1 and 2, show the impact of time and space harmonics on the initial steady state conditions of the synchronous generator for different time harmonic levels. These initial condition values have influence on the system eigen-values and hence affect dynamic behavior of the SMIB system including SSSC. Finally, the eigen-values of the system are calculated using equation 16 for different (time and space) harmonic levels and the results are shown in Table 3. The results demonstrate that harmonic phase angles have considerable influence on the steady-state and dynamic performance of synchronous generator and power system. According to Table 3, by increasing the level of the harmonic distortions, considerable variations in the system eigen-values are observed. The influence of phase angle can not be neglected as it might cause instability of the system by shifting the eigen-value to the right hand side of the imaginary axis.

Table 1. Impact of space and time harmonics on the induced rotor current harmonics and load angle

Space Harmonics = 2× Base Value

Time Harmonic Level Load Angle (δo) [% Deviation]

Base* ( 0)7(i =θ ) 38.70o [-1.02%**]

Base ( 2/)7( πθ =i) 39.56o [+1.18%]

2× Base ( 0)7(i =θ ) 38.71o [-0.9%]

2× Base( 2/)7( πθ =i) 40.33o [+3.15%]

*)0.3pu, 0.5pu, 0.2pu and 0.1pu for the 3rd to 9th harmonics, respectively. **) Percentage of deviation from the base value without harmonics.

Table 2. Impact of space and time harmonics on the steady-state operating conditions

Space Harmonics = 2× Base Value

Time Harmonic Level Stator Current (on d-q axis)

Stator fluxes (on d-q axis)

Base* ( 0)7(i =θ )

4297.0

9030.0

0

0

==

q

d

I

I

6283.0

9081.0

0

0

−==

q

d

ψψ

Base ( 2/)7( πθ =i)

4161.0

9094.0

0

0

==

q

d

I

I 6391.0

9672.0

0

0

−==

q

d

ψψ

2× Base ( 0)7(i =θ )

4297.0

9030.0

0

0

==

q

d

I

I 6278.0

8337.0

0

0

−==

q

d

ψψ

2× Base( 2/)7( πθ =i)

4038.0

9149.0

0

0

==

q

d

I

I 6501.0

9395.0

0

0

−==

q

d

ψψ

Table 3. Impact of space and time harmonics on eigen-values of SMIB system (Figure 1)

Space Harmonics = 2× Base Value Time Harmonic Level System Eigen-Values Stability

Base* ( 0)7(i =θ )

0.0160j 0.0240-

,5.4024j 0.1301-

±± Stable

Base ( 2/)7( πθ =i)

0.0217j 0.0179

4.7428 0.1981-

±+± j Unstable

2× Base ( 0)7(i =θ )

0.0160j 0.0259-

,5.4105j 0.1282-

±± Stable

2× Base( 2/)7( πθ =i)

0.0039

0.3738

3.3724 0.4181-

++

± j Unstable

5. CONCLUSION

Small-signal stability of distorted Single Machine Infinite Bus System installed with SSSC is investigated using the eigen-value analysis. A nonlinear model of synchronous generator suitable for harmonic studies is used to compute the modified fundamental dq components of the stator flux including additional terms due to the presence of time and space harmonics. The generator initial conditions and system state space equations are computed and used to analyze the system dynamic stability. Simulation results for two operating conditions are presented. The main contributions are:

• Time and space harmonics introduce new additional terms in the system initial condition equations (Eqs.3 and 9).

• Harmonic phase angles have influence on the steady-state performance of synchronous generator (Table 2).

• Time and space harmonics change the elements of the system state matrix (Eq.16 )

• Time and space harmonics will change the system eigen-values and hence affect the system dynamic behavior (Table 3).

• Harmonic phase angles have considerable impact on steady state and dynamic behavior of the system and can cause power system instability (Tables 1-3).

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APPENDIX A SPECIFICATIONS OF SYSTEM PARMETERS

AND LOADING CONDITIONS Synchronous machine parameters in per unit of machine rating (555MVA, 24kV, 0.9p.f, 60Hz, 3600RPM turbine generator) are:

Stator Parameters: Ls=1.060,Lm=0.0140,Ms=0.4550,Ra =0.003.

Rotor Parameters: RF =0.0006, LFd = 1.551.

Stator Rotor Mutual Parameter: MF = 1.386.

System and SSSC parameters: Xtotal = 0.7, RE = 0.

=DCC 1, =DCoV 1, k=1.